2009年安徽高考高中數(shù)學(xué)基礎(chǔ)知識歸納

第一部分   集合

1.理解集合中元素的意義是解決集合問題的關(guān)鍵：元素是函數(shù)關(guān)系中自變量的取值？還是因變量的取值？還是曲線上的點？…

2 .數(shù)形結(jié)合是解集合問題的常用方法：解題時要盡可能地借助數(shù)軸、直角坐標(biāo)系或韋恩圖等工具，將抽象的代數(shù)問題具體化、形象化、直觀化，然后利用數(shù)形結(jié)合的思想方法解決

3.(1) 元素與集合的關(guān)系：,.

（2）德摩根公式： .

（3）       注意：討論的時候不要遺忘了的情況.

（4）集合的子集個數(shù)共有 個；真子集有?1個；非空子集有 ?1個；

非空真子集有?2個.

4．是任何集合的子集，是任何非空集合的真子集.

第二部分  函數(shù)與導(dǎo)數(shù)

1．映射：注意: ①第一個集合中的元素必須有象；②一對一或多對一.

2．函數(shù)值域的求法：①分析法 ；②配方法 ；③判別式法 ；④利用函數(shù)單調(diào)性 ；⑤換元法 ；

⑥利用均值不等式 ； ⑦利用數(shù)形結(jié)合或幾何意義（斜率、距離、

絕對值的意義等）；⑧利用函數(shù)有界性（、、等）；⑨平方法；⑩ 導(dǎo)數(shù)法

3．復(fù)合函數(shù)的有關(guān)問題:

（1）復(fù)合函數(shù)定義域求法：

① 若f(x)的定義域為［a，b］,則復(fù)合函數(shù)f[g(x)]的定義域由不等式a ≤  g(x) ≤  b解出

② 若f[g(x)]的定義域為[a,b],求 f(x)的定義域，相當(dāng)于x∈[a,b]時，求g(x)的值域.

（2）復(fù)合函數(shù)單調(diào)性的判定：

①首先將原函數(shù)分解為基本函數(shù)：內(nèi)函數(shù)與外函數(shù)

②分別研究內(nèi)、外函數(shù)在各自定義域內(nèi)的單調(diào)性

③根據(jù)“同性則增，異性則減”來判斷原函數(shù)在其定義域內(nèi)的單調(diào)性.

4．分段函數(shù)：值域（最值）、單調(diào)性、圖象等問題，先分段解決，再下結(jié)論。

5．函數(shù)的奇偶性:

⑴函數(shù)的定義域關(guān)于原點對稱是函數(shù)具有奇偶性的必要條件

⑵是奇函數(shù)；是偶函數(shù)_.

⑶奇函數(shù)在0處有定義，則

⑷在關(guān)于原點對稱的單調(diào)區(qū)間內(nèi)：奇函數(shù)有相同的單調(diào)性，偶函數(shù)有相反的單調(diào)性

⑸若所給函數(shù)的解析式較為復(fù)雜，應(yīng)先等價變形，再判斷其奇偶性

6．函數(shù)的單調(diào)性:

⑴單調(diào)性的定義：

①在區(qū)間上是增函數(shù)當(dāng)時有；

②在區(qū)間上是減函數(shù)當(dāng)時有；

⑵單調(diào)性的判定：①定義法：一般要將式子化為幾個因式作積或作商的形式，以利于判斷符號；②導(dǎo)數(shù)法（見導(dǎo)數(shù)部分）；③復(fù)合函數(shù)法；④圖像法

注：證明單調(diào)性主要用定義法和導(dǎo)數(shù)法。

7．函數(shù)的周期性：

(1)周期性的定義：對定義域內(nèi)的任意，若有 （其中為非零常數(shù)），則稱函數(shù)為周期函數(shù)，為它的一個周期。所有正周期中最小的稱為函數(shù)的最小正周期。如沒有特別說明，遇到的周期都指最小正周期。

（2）三角函數(shù)的周期：① ；② ；③；

④ ；⑤

(3)與周期有關(guān)的結(jié)論：

或 的周期為

8．基本初等函數(shù)的圖像與性質(zhì)：

㈠.⑴指數(shù)函數(shù)：；⑵對數(shù)函數(shù):；

⑶冪函數(shù)： （ ；⑷正弦函數(shù):；⑸余弦函數(shù)： ；

（6）正切函數(shù)：；⑺一元二次函數(shù)：（a≠0）；⑻其它常用函數(shù)：

①     正比例函數(shù)：；②反比例函數(shù)：；③函數(shù)

㈡.⑴分?jǐn)?shù)指數(shù)冪：；（以上，且）.

⑵.①；       ②；

③；  ④.

⑶.對數(shù)的換底公式:.對數(shù)恒等式:.

9．二次函數(shù)：

⑴解析式：①一般式：；②頂點式：，為頂點；

③零點式： （a≠0）.

⑵二次函數(shù)問題解決需考慮的因素：

①開口方向；②對稱軸；③端點值；④與坐標(biāo)軸交點；⑤判別式；⑥兩根符號。

二次函數(shù)的圖象的對稱軸方程是，頂點坐標(biāo)是。

10．函數(shù)圖象：

⑴圖象作法 ：①描點法 （特別注意三角函數(shù)的五點作圖）②圖象變換法 ③導(dǎo)數(shù)法

⑵圖象變換：

①     平移變換：?)，―――左“+”右“－”；

              ?) ―――上“+”下“－”；

②     對稱變換：?)；?)；

?) ； ?)；

③     翻折變換：

?)―――（去左翻右）y軸右不動，右向左翻（在左側(cè)圖象去掉）；

?)―――（留上翻下）x軸上不動，下向上翻（||在下面無圖象）；

11．函數(shù)圖象（曲線）對稱性的證明：

(1)證明函數(shù)圖像的對稱性，即證明圖像上任意點關(guān)于對稱中心（對稱軸）的對稱點仍在圖像上；

（2）證明函數(shù)與圖象的對稱性，即證明圖象上任意點關(guān)于對稱中心（對稱軸）的對稱點在的圖象上，反之亦然。

注：①曲線C₁:f(x,y)=0關(guān)于原點（0,0）的對稱曲線C₂方程為：f(－x,－y)=0;

曲線C₁:f(x,y)=0關(guān)于直線x=0的對稱曲線C₂方程為：f(－x, y)=0;

曲線C₁:f(x,y)=0關(guān)于直線y=0的對稱曲線C₂方程為：f(x, －y)=0;

曲線C₁:f(x,y)=0關(guān)于直線y=x的對稱曲線C₂方程為：f(y, x)=0

②f(a+x)=f(b－x) （x∈R）y=f(x)圖像關(guān)于直線x=對稱；

特別地：f(a+x)=f(a－x) （x∈R）y=f(x)圖像關(guān)于直線x=a對稱.

③的圖象關(guān)于點對稱.

特別地：的圖象關(guān)于點對稱.

④函數(shù)與函數(shù)的圖象關(guān)于直線對稱;

      函數(shù)與函數(shù)的圖象關(guān)于直線對稱。

12．函數(shù)零點的求法：

⑴直接法（求的根）；⑵圖象法；⑶二分法.

(4)零點定理：若y=f(x)在[a,b]上滿足f(a)?f(b)<0 , 則y=f(x)在（a,b)內(nèi)至少有一個零點。

13．導(dǎo)數(shù)：

⑴導(dǎo)數(shù)定義：f(x)在點x₀處的導(dǎo)數(shù)記作

⑵常見函數(shù)的導(dǎo)數(shù)公式: ①；②；③；④；⑤；⑥；⑦；⑧ 。

⑶導(dǎo)數(shù)的四則運(yùn)算法則：

⑷（理科）復(fù)合函數(shù)的導(dǎo)數(shù)：

⑸導(dǎo)數(shù)的應(yīng)用：                                                      

①利用導(dǎo)數(shù)求切線：注意：?)所給點是切點嗎？?)所求的是“在”還是“過”該點的切線？

②利用導(dǎo)數(shù)判斷函數(shù)單調(diào)性：i）是增函數(shù)；ii）為減函數(shù)；iii）為常數(shù)；

③利用導(dǎo)數(shù)求極值：?）求導(dǎo)數(shù)；?）求方程的根；?）列表得極值。

④利用導(dǎo)數(shù)求最大值與最小值：?）求極值；?）求區(qū)間端點值（如果有）；?）比較得最值。

第三部分  三角函數(shù)、三角恒等變換與解三角形

1．⑴角度制與弧度制的互化：弧度，弧度，弧度

⑵弧長公式：；扇形面積公式：。

2．三角函數(shù)定義:角終邊上任一點（非原點）P,設(shè) 則：

3．三角函數(shù)符號規(guī)律：一全正，二正弦，三正切，四余弦；（簡記為“全s t c”）

4．誘導(dǎo)公式記憶規(guī)律：“奇變偶不變，符號看象限”

5．⑴ 對稱軸：令，得  對稱中心：；

⑵ 對稱軸：令，得；對稱中心：；

⑶周期公式:①函數(shù)及的周期 (A、ω、為常數(shù)，

且A≠0).②函數(shù)的周期 (A、ω、為常數(shù)，且A≠0).

6．同角三角函數(shù)的基本關(guān)系：

7．三角函數(shù)的單調(diào)區(qū)間及對稱性：

  ⑴的單調(diào)遞增區(qū)間為,單調(diào)遞減區(qū)間為

，對稱軸為,對稱中心為.

⑵的單調(diào)遞增區(qū)間為,單調(diào)遞減區(qū)間為，

對稱軸為,對稱中心為.

⑶的單調(diào)遞增區(qū)間為，對稱中心為.

8．兩角和與差的正弦、余弦、正切公式：

①；；

.

②；.

③=(其中,輔助角所在象限由點所在的象限

決定, ).

9．二倍角公式：①.

②（升冪公式）.

（降冪公式）.

10．正、余弦定理：

⑴正弦定理：  （是外接圓直徑 ）

注：①；②；③。

⑵余弦定理：等三個； 等三個。

11.幾個公式:⑴三角形面積公式：①（分別表示a、b、c邊上的高）；②.③

⑵內(nèi)切圓半徑r=； 外接圓直徑2R=

第四部分   立體幾何

1．三視圖與直觀圖：⑴畫三視圖要求：正視圖與俯視圖長對正；正視圖與側(cè)視圖高平齊；側(cè)視圖與俯視圖寬相等。 ⑵斜二測畫法畫水平放置幾何體的直觀圖的要領(lǐng)。

2．表（側(cè)）面積與體積公式：

⑴柱體：①表面積：S=S_側(cè)+2S_底；②側(cè)面積：S_側(cè)=；③體積：V=S_底h

⑵錐體：①表面積：S=S_側(cè)+S_底；②側(cè)面積：S_側(cè)=；③體積：V=S_底h：

⑶臺體：①表面積：S=S_側(cè)+S_下底;②側(cè)面積：S_側(cè)=;③體積：V=（S+）h；

⑷球體：①表面積：S=；②體積：V=  .

3．位置關(guān)系的證明（主要方法）：

⑴直線與直線平行：①公理4；②線面平行的性質(zhì)定理；③面面平行的性質(zhì)定理。

⑵直線與平面平行：①線面平行的判定定理；②面面平行線面平行。

⑶平面與平面平行：①面面平行的判定定理及推論；②垂直于同一直線的兩平面平行。

⑷直線與平面垂直：①直線與平面垂直的判定定理；②面面垂直的性質(zhì)定理。

⑸平面與平面垂直：①定義----兩平面所成二面角為直角；②面面垂直的判定定理。

注：以上理科還可用向量法。

4.求角：（步驟-------Ⅰ.找或作角；Ⅱ.求角）

⑴異面直線所成角的求法：

①平移法：平移直線，構(gòu)造三角形；②用向量法

⑵直線與平面所成的角：

①直接法（利用線面角定義）；②用向量法

5.求距離：（步驟-------Ⅰ.找或作垂線段；Ⅱ.求距離）

點到平面的距離：①等體積法；②向量法

6．結(jié)論：

⑴棱錐的平行截面的性質(zhì)如果棱錐被平行于底面的平面所截，那么所得的截面與底面相似，截面面積與底面面積的比等于頂點到截面距離與棱錐高的平方比（對應(yīng)角相等，對應(yīng)邊對應(yīng)成比例的多邊形是相似多邊形，相似多邊形面積的比等于對應(yīng)邊的比的平方）；相應(yīng)小棱錐與小棱錐的側(cè)面積的比等于頂點到截面距離與棱錐高的平方比．

⑵長方體從一個頂點出發(fā)的三條棱長分別為a，b，c，則體對角線長為，全面積為2ab+2bc+2ca，體積V=abc。

⑶正方體的棱長為a，則體對角線長為，全面積為，體積V=。

⑷球與長方體的組合體: 長方體的外接球的直徑是長方體的體對角線長.

  球與正方體的組合體:正方體的內(nèi)切球的直徑是正方體的棱長, 正方體的棱切球的直徑是正方體的面對角線長, 正方體的外接球的直徑是正方體的體對角線長.

⑷正四面體的性質(zhì)：設(shè)棱長為，則正四面體的：

①     高：；②對棱間距離：；③內(nèi)切球半徑：；④外接球半徑：。

第五部分   直線與圓

1．斜率公式：，其中、.

直線的方向向量，則直線的斜率為=.

2.直線方程的五種形式：

(1)點斜式： (直線過點，且斜率為)．

(2)斜截式：

08-09學(xué)年度高二第二學(xué)期3月份月考數(shù)學(xué)試卷

試題詳情

冠龍高級中學(xué)

2008―2009學(xué)年度第二學(xué)期高三年級月考

英 語 試 題

本試卷分為第I卷和第II卷兩部分，全卷滿分150分。考試時間120分鐘。

第Ⅰ卷（共105分）

I. Listening Comprehension  （30%）

Part A  Short Conversation                     

Directions: In Part A. you will hear ten short conversations between two speakers. At the end of each conversation, a question will be asked about what was said. The conversations and the questions will be spoken only once. After you hear a conversation and the question about it, read the four possible answers on your paper, and decide which one is the best answer to the question you have heard.

1. A. The desk drawer won’t open                      B. The pen is out of ink

C. She cannot find her pen                             D. She is angry with the man

2. A. It’s exactly 5:35                                       B. It’s a little before 5:30

  C. It’s exactly 5:30                                        D. It’s a little after 5:30.

3. A. Where he is studying                                B. What subject he is studying

  C. How long he has been in Paris                   D. When he is returning

4. A. Yes, because he has plenty of money           B. No, because it’s not for sale

  C. Yes, if he borrows money from the woman  D. No, he wants to wait until it goes on sale

5. A. It’s boring                                               B. The subject is hard

  C. It is hard to understand teacher’s speech      D. It is hard to pronounce some words

6. A. The man was waiting to do something        B. The man was asking for directions

  C. The man was making a trip                        D. The man was asking for help

7. A. Take the bus with him                               B. Drive him home from the station

  C. Drive him to work                                    D. Drive him to the station

8. A. He is tired of using his new computer        

B. He has just assembled the computer

  C. He has used his new computer for waiting his report

  D. He has found something wrong with the new computer

9. A. Shopping for new clothes                          B. Having a party

  C. Looking for her black dress                       D. Getting dressed

10. A. She does not like the apartment.

   B. She is concerned about the rent for the apartment.

   C. She wants to live in the apartment

   D. She agrees with the man.

Part B Passages

Directions: In Part B, you will hear two short passages, and you will be asked three questions on each of the passages. The passages will be read twice, but the questions will be spoken only once. When you hear a question, read the four possible answers on your paper and decide which one would be the best answer to the question you have heard.

Questions 11 through 13 are based on the following passage.

11. A. Right decisions                                B. Wrong decisions

   C. Experience                                      D. Opportunity

12. A. She was lucky to have many marvelous opportunities.

   B. She was afraid of taking risks.

   C. She made wrong decisions in her work.

   D. She felt bad when she made a mistake.

13. A. Forget it and move on,

   B. Don’t make the same mistake repeatedly.

   C. Mistakes are a necessary part of learning

   D. The more mistakes you make, the better you will become.

Questions 14 through 16 are based on the following news.

14. A. The third            B. The fourth                C. The fifth           D. The sixth

15. A. He could not see their eyes                      B. He thought they were useless

   C. He could see no willingness to help           D. He thought they were in a hurry

16. A. He was grateful   B. He was curious         C. He was happy    D. He was excited

Part C Longer Conversations

Directions: In Part C, you will hear two longer conversations. The conversations will be read twice. After you hear each conversation, you are required to fill in the numbered blanks with information you have heard. Write your answers on your answer sheet.

Questions 17 through 20 are based on the following conversation.

Complete the following table using NO MORE THAN TWO WORDS for each question.

Linda’s Dairy

                                                  Dec.28^th.,2002

Today’s the second day of our camping holiday.

Got up    At (17) _____________

Problem   (18) ________________

Solution   Put (19) ______________on our camping car to attract (20) _______.

          The noise they made woke us up.

It did work. What a brilliant idea!

Questions 21 through 24 are based on the following conversation.

Complete the following table using NO MORE THAN THREE WORDS for each question.

BIKES FOR HIRE

Kinds of bikes                      Push bikes and (21)_________.

Minimum age                      Must be (22) _____________.

Minimum days                     At least (23) ______________.

Can take kids on the back?           Yes.

Necessity for booking                (24)_____________________.

II. Grammar and Vocabulary 16%

Directions: Beneath each of the following sentences there are four choices marked A, B. C and D.   Choose the one answer that best completes the sentence.

25. The company had about 20 notebook computers but only one-third ____ used regularly. Now we have 60 working all day long．

A. is                            B. are                           C. was                          D. were

26. I have just finished a book about a piano player ____ sole ambition in life was to perform for the pope.

A. whose               B. which                      C. who                         D. whom

27. _____ well prepared you are, you still need a lot of luck in mountain climbing.

A. No mater          B. Whatever                 C. However                  D. Although

28. We are aware that, ____, the situation will get worse.

A. if not dealing with carefully                   B. if dealt not carefully with

C. if not carefully dealt with                       D. if not carefully dealing with

29. _______ snacks and drinks, but they also brought cards for entertainment when they had a picnic in the forest.

A. Not only they brought                           B. Not only did they bring

C. Not only brought they                           D. Not only they did bring

30. ---English has a large vocabulary, hasn’t it?

   ---Yes. ____ more words and expressions and you will find it easier to read and communicate.

A. Know               B. Knowing                  C. To know                   D. Known

31. It was the university ___ he was studying ____ sent him abroad.

A. which, that        B. where, that               C. that, which               D. that, had

32. ___ electricity, you should turn off the lights when you leave a room.

A. Saving              B. Having saved            C. Being saved              D. To save

33. Wolves will soon be extinct everywhere ____ we stop killing them.

A. because             B. if               C. unless               D. when

34. Go and get your coat. It’s ___ you left it.

A. there                 B. where               C. there where                     D. where there

35. Many thousands of people were walking along the Bund on the evening of our National Day, ____ the magnificent night scene.

A. to enjoy            B. enjoying            C. to watch                   D. noticing

36. ____ he made an important speech at yesterday’s meeting was true.

A. That                 B. Why                 C. It was said                D. When

37. I would appreciate ___ it a secret.

A. your keeping     B. that you keep     C. you to keep                     D. that you will keep

38. It is reported that 82% of women ___ 50% or more of their income on clothing were aged between 25 to 35.

A. spent                B. to spend            C. spend                       D. spending

39. She had never spent a ___ day.

A. more worry              B. most worrying          C. more worrying          D. most worried

40. With time going on, the method she has stuck ___ very effective and it was worthy of ___.

A. proves, adopting                                   B. to proving, being adopted

C. to proved, being adopted                 D. to prove, adopting

III. Vocabulary 9%

Directions: Complete the passage with the words in the box. There is one extra word.

The new figures are the result of detailed research that gives the most accurate assessment of the disease that kills at least a million people a year. Scientists now believe there are about 515 million _1__ of malaria (瘧疾) out of 2.2 billion people who are at risk ----- about a third of the world’s _2__.

Malaria has never captured the public imagination as AIDS has done, even though children are its chief _3___. Malaria is old and AIDS is new. Most important, malaria is not a disease that __4__ the West ---- except for those fortunate enough to holiday in the tropics ---- while AIDS _5__ us all.

Six years ago the WHO set a _6__ to halve the number of deaths by the year 2010, but instead the number has risen by at least a quarter, and in some areas by as much as 50 percent, _7__ victims have not had the right drugs. Hundreds of thousands of children have died needlessly and the disease has gone virtually unnoticed in the West. Malaria has been a cause of great _8__ for humanity since ancient times and although it is largely preventable with the use of mosquito nets and insecticides, it remains one of the biggest __9_ of children under five.

IV. Cloze (15%)

Directions: For each blank in the following passages there are four words or phrases marked A, B, C and D. Fill in each blank with the word or phrase that best fits the context.

Crime has its own cycle, a magazine reported some years ago. Police records that were studied for five years from over 2,400 cities and towns show a surprising _41_ between changes in the season and crime patterns.

The pattern of crime has varied very little over a long period of years. Murder _42_ its high during July and August, as do rape and other __43__ attacks. Murder, __44__, is more than seasonal: it is a weekend crime. It is also a high time crime: 62 percent of murders are __45__ between 6 pm and 6 am.

Unlike the summer high in crimes of bodily harm, burglary has a(n) __46__ cycle. You are most __47__ to be robbed between 6 p.m. and 2 a.m. __48__ a Saturday night in December, January or February. May is the most non-criminal month of all except for one strange statistics.   __49__ dog bites are reported in this month than in any other month of the year.

Apparently our intellectual seasonal cycles are __50__ different from our criminal tendencies. Professor Huntington, of the Foundation for the Study of Cycles, made extensive __51__ to discover the seasons when people read serious books, attend scientific meetings, make __52__ scores on examinations and propose the most changes to parents. In all instances, he found a spring peak and an autumn peak __53__ by a summer low. On the other hand, Professor Huntington’s studies indicated that June is the __54__ month for suicides and __55__ to mental hospitals. June is also a peak month for marriages!

41．A．difference        B．link                 C．coincidence      D．similarity

42．A．reaches            B．increases          C．kills                D．falls